_ 1 ^
N = n(I - A) - F [2]
where n and F contain the sectoral employment coefficients and final demands as in [1]
expressed as diagonal matrices. N is now the square sector-by-sector matrix where the ij-th
element Nij is employment in (row) sector i generated by final demand in sector j. The row-sum
of each i-th row of N gives the employment generated within sector i to supply its output for
final use and to all intermediate users. This is the sectoral allocation of employment as
conventionally measured. Each j-th column-sum of N gives the employment generated economy-
wide for the production of the j-th sector’s final demand. This is the VIS allocation, attributing
employment to the sector of the final demand which it serves, independent of the sector in
which it is located. This model is developed more fully in Appendix 2.
The allocation of employment across vertically integrated sectors is illustrated with a numerical
example in Table A1. The top panel shows the input-output flows for a three-sector economy in
conventional format. Employment in each sector is given in the far right column, showing sector
3 as much the largest employer. Below are the Leontief (I - A) and inverse B = (I - A)-1
coefficient matrices. In the panel below the final demand and employment vectors are
converted to diagonal matrices, and the matrix calculation B * F completed, giving the total
amount of output required to sustain the given levels of final demand. The bottom panel
presents the calculation of the matrix n^* B * F where the quantity of output B * F is pre-
multiplied by the diagonal matrix n of employment coefficients to give the level of employment
required in each sector to sustain the given vector of final demands. Reading along the rows, this
matrix shows the level of employment required in the row industry to support each successive
element in the (column) vector of final demands; industry 1 employs 7.77 workers to support
final demand for its own output, 1.59 to supply intermediate inputs into the final output of
industry 2, and 0.63 towards final demand in industry 3. Total employment in industry 1,
supporting the final demands from all three industries, is 10 workers. The row sums of the
employment matrix give back the initial within-sector employment levels. Reading down column
1 traces the employment required in sectors 1, 2 and 3 to sustain final demand for sector 1’s
output - 7.77, 3.60 and 27.18 units respectively. The column total of 38.56 is the employment
required economy-wide to produce the 110 units of final output of sector 1. This is the
employment attributed to industry 1 on a VIS basis. Summing across the column totals for the
vertically integrated sectors returns the economy’s total employment of 110 units.